Numerical accuracy conceptions of past music theorists

Hello

I have a thought which I'll put down quickly. I
believe it is constructive. Many great thinkers of
the past believed in an order in nature which today
seems naive. Thus Johannes Kepler proposed that
the distances of planets from the sun corresponded
to the sizes of a set of symmetrical solids - icosahedron
dodecahedron, cube, etc. which could be inscribed
within each other, and then later dropped this idea.
Isaac Newton spent much time trying to calculate
the exact time to the creation of the world - down
to the day and hour. I believe that musical theorists
also had the belief that there was an underlying
"perfect" scheme of some kind laid down by the
creator which, if they could only find it, lay at the
root of musical art as well as everything else. The
writings of some theorists seem to be elaborate and
detailed to a degree beyond which would seem today
reasonably possible to expect. In musical performance
especially with non-fixed pitch instruments and voice,
it seems unlikely that - with a few unusual exceptions -
pitch accuracies closer than to a cent or a few cents
could be achieved consistently in practical performances.
However, some theoreticians seem to have carried out
ideal calculations to limits of accuracy beyond this.

I believe that most musicians from before the time
when equal temperament began to be forcefully promoted
in the 19th century uncritically assumed that there was
some kind of underlying order of nature underlying musical
pitch - note frequencies - which was based on numbers
of the integer/small whole number kind. Then, because
approximations to these - mean tone temperament and later
equal temperament yielded performances which sounded
not completely different than would just performances, the
belief in the need to hold absolutely strictly to small whole
number ratios in music began to be shattered.

A spokesman for the new view is the musical mathematician
W. S. B. Woolhouse, who wrote in the 19th century: "It is very
misleading to suppose that the necessity of temperament applies
only to instruments which have fixed tones. Singers and performers
on perfect instruments must all temper their intervals, or they
could not keep in tune with each other, or even with themselves;
an on arriving at the same notes by different routes, would be
continually finding a want of agreement. The scale of equal
temperament obviates all such inconveniences, and continues to
be universally accepted with unqualified satisfaction by the most
eminent vocalists; and equally so by the most renowned and
accomplished performers on stringed instruments, although these
instruments are capable of an indefinite variety of intonation.
The high development of modern instrumental music would not
have been possible, and could not have been acquired, without
the manifold advantages of tempered intonation by equal semitones,
and it has, in consequence, long become the established basis of
tuning."

At present, although most of us see the belief in the absolute
supremacy of integers as applied to music to be naive, it also
seems, as Paul Erlich has said with regard to simultaneous notes,
that integer ratios or very near integer ratios (closer than called
for by 12-EQT) correspond to a psychological reality as regards
our perception of music.

I believe that Mr. Woolhouse, in stating that equal temperament
is rightly the long established basis of tuning, really went too far
and overlooked the fact of experience that the deviations of equal
temperament from the just ratios are so large that they really
do have an appreciable effect on the sound of music performed in
equal temperament.

Some of Mr. Woolhouse's contemporaries strongly disagreed
with him, too.

This thought may be incomplete, but I hope it is a constructive
contribution.

Dave Hill, La Mesa, CA

Ascend11@aol.com wrote:

> At present, although most of us see the belief in the absolute
> supremacy of integers as applied to music to be naive, it also
> seems, as Paul Erlich has said with regard to simultaneous notes,
> that integer ratios or very near integer ratios (closer than called
> for by 12-EQT) correspond to a psychological reality as regards
> our perception of music.

I am not quite sure what you mean in the 1st phrase here. It is interesting
that those composers of notoriety who use different tunings, have use a JI
system. Partch, Young, Harrison, and Riley.

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

[Paul Erlich wrote...]
>Whether or not Mr. Woolhouse went "too far", Mr. Hill has failed to
address >Woolhouse's point here. Furthermore, far from overlooking the
errors from >just intonation, Woolhouse sought the best way to reduce them
while >preserving the musical meaning of the notes in the Western tradition.

This would unquestionably involve meantone. But there are two points I'd
like to make here:

[1.] Free-pitched performances in the western tradition do certainly _not_
perform in temperament, if temperament is a _consistent_ detuning of just
ratios for the purpose of reducing the size of a pitch set. In the case of
brass quintets and Barbershop quartets, and other a capella groups,
surprisingly good adaptive JI is achieved by the most renowned performers,
and the emphasis is on vertical sonorities over melodic coherence
(JI-related melodic artifacts can be heard in some of the best performances).

Even in the case of bare melody temperament is not used (or melody with a
relatively quiet and rhythmically un-related accompaniment). I challenge
anyone to find a performance with melodic fifths consistently tuned 6 cents
flat. As with vertical sonorities, JI must be applied locally to melody to
make sense (some theorists never seem to tire of pointing out that global
"fixed" JI doesn't make sense harmonically). The intervals between melodic
notes tend towards ratios of small whole numbers for the same reason that
harmonic intervals do -- the hearing system likes to fit things into a
harmonic series. The accuracy is far less than with harmonic intervals, at
least because the roughness mechanism is not involved (so melodies produced
on a tempered instrument will sound okay).

On the global scale, there is a desire to simplify the Rothenberg
difference matrix. Pythagorean chains, MOS, constant structures, and
tetrachordality all simplify the difference matrix. I reject the idea that
only ratios of 3 are important in melodic tuning. Often, the desire to
simplify the DM makes the global scale definable in terms of a single just
interval, and this interval has historically been the 3/2, because 3/2 is
such a strong consonance. But chains of any strong consonance could work,
and even in 3-based scales, the approximation is only a global one.
Performers often bend notes locally so that melodic (as well as harmonic)
intervals can be closer to 5/4, 7/4, etc, and in isolated runs one can even
hear stuff like :10, :11, :12.

[2.] As concerns new music, the "problems" of JI would naturally be used
to advantage by the composer. Music exists, still very much in the western
tradition, which demonstrates this.

[Kraig Grady wrote...]
>I am not quite sure what you mean in the 1st phrase here. It is interesting
>that those composers of notoriety who use different tunings, have use a JI
>system. Partch, Young, Harrison, and Riley.

Let's not forget Wyschnegradsky, Carillo, Haba, Mandelbaum, Blackwood, or
Haverstick!

-Carl

Carl!
Despite the worth of the following , they are not as well known!

Carl Lumma wrote:

>
>
> [Kraig Grady wrote...]
> >I am not quite sure what you mean in the 1st phrase here. It is interesting
> >that those composers of notoriety who use different tunings, have use a JI
> >system. Partch, Young, Harrison, and Riley.
>
> Let's not forget Wyschnegradsky, Carillo, Haba, Mandelbaum, Blackwood, or
> Haverstick!
>
> -Carl
>
>

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

>And in fact, for ratios of 5 or higher, the accuracy with which ratios of
>_melodic_ intervals are reckoned is so low that factors of "custom" or
>"familiarity" are far more important than the harmonic series.

That's hard to determine. And what are the driving forces behind custom,
then? I'd call +/-10 cents a fair estimate of the accuracy of singing the
"base-two under" 7-limit melodic intervals, which is closer than 19-tone
equal temperament.

Far greater accuracy is possible when a drone is present. Granted, these
would be "harmonic" intervals by our current convention, but... clearly a
melody over a drone is still a melody in some important ways (otherwise we
would not find approximations to the linear series in Indian and Persian
melodies).

[Reference Bobby McFerrin's "The Voice"]

>Furthermore, if your first sentence above is true, why are minor arpeggios
>at least as common as major arpeggios in non-harmonized music from around
>the world (at least what I've heard)?

You're assuming they approximate a utonal structure? Stuff I've heard is
usually closer to 7/6 or 19/16. That may be because of the linear series,
or the harmonic series, but in neither case are subharmonics implicated.
There is the exception of ethnic flutes, which often play subharmonic
series melodies, and after hearing such, it isn't too hard to sing them...

>>and this interval has historically been the 3/2, because 3/2 is such a
>>strong consonance. But chains of any strong consonance could work,
>
>Speaking again strictly of melodic tuning, I fail to see any evidence of
>that.

Historical evidence, maybe not (although Wilson may disagree). But
certainly the melodic validity of Keenan's chain-of-minor-thirds scales --
or the 22-out-of-41 scale you proposed -- is not in dispute, to say nothing
of the myriad of proper MOS's of the 5/4, 7/6, 7/5, and 7/4.

-Carl

>>That's hard to determine. And what are the driving forces behind custom,
>>then?
>
>I don't know; join the sociology list.

They are either "frozen accidents" or something that can be discussed on
this list.

>>I'd call +/-10 cents a fair estimate of the accuracy of singing the
>>"base-two under" 7-limit melodic intervals, which is closer than 19-tone
>>equal temperament.
>
>Come again?

7/4, 5/4, 3/2

>>[Reference Bobby McFerrin's "The Voice"]
>
>That's a great album -- which track are you thinking of?

Harmonic series segment like things can be heard throughout the album,
especially on track 7 in the "We're in the money" section.

>>You're assuming they approximate a utonal structure? Stuff I've heard is
>>usually closer to 7/6 or 19/16. That may be because of the linear series,
>>or the harmonic series, but in neither case are subharmonics implicated.
>>There is the exception of ethnic flutes, which often play subharmonic
>>series melodies, and after hearing such, it isn't too hard to sing them...
>
>Are you buying into the Schlesinger equally-spaced holes argument? No,
>flutes aren't quite like strings.

What do equal-spaced holes on flutes produce, then?

>I think they could arise on other instruments and even vocally, though; if
>the common overtone happens to coincide with a resonance, a subharmonic
>arpeggio would really stand out.

That's another thing -- perhaps subharmonic structures are important in
melody, assuming that first sentence of mine was true. But maybe it's true
not because of the virtual pitch mechanism, but because of the spectral
mechanism -- maybe the ear just hears just melodic intervals better because
it is aware of a common harmonic.

>By the way, Carl, you've referred to "linear series" twice now. Are you
>being Wilsonified?

I was Wilsonified long ago.

>>Historical evidence, maybe not (although Wilson may disagree).
>
>Any examples?

Not that I can remember, but I seem to recall him explaining an ethnic
scale as a chain of something other than 3/2's (was it one of the
permutations articles?). Kraig?

>>But certainly the melodic validity of Keenan's chain-of-minor-thirds scales
>> -- or the 22-out-of-41 scale you proposed -- is not in dispute,
>
>I recall much dispute over the latter from yourself, and please give us an
>mp3 of a nice melody with the former.

I disputed the scale only because it's so big -- most melodies would be
subsets of it. And due to perceptual limitations, even a melody that did
use the entire scale would probably fail to convey a 22-member melodic
pitch set. But it surely contains a wealth of great subsets.

Eeek! I've been having the worst time getting to my server today. My ISP
is periodically taken with loosing connection to whole portions of IP
space. At last, you may try the following, in the proper 8-tone subset of
Keenan's scale...

http://lumma.org/01.mp3 - 274K
http://lumma.org/02.mp3 - 634K
http://lumma.org/03.mp3 - 574K
http://lumma.org/04.mp3 - 1.02M

..or, go for all of them, plus 3 more at...

http://lumma.org/8all.zip - 3.37M

-Carl

>>That's hard to determine. And what are the driving forces behind custom,
>>then?
>
>I don't know; join the sociology list.

They are either "frozen accidents" or something that can be discussed on
this list.

>>I'd call +/-10 cents a fair estimate of the accuracy of singing the
>>"base-two under" 7-limit melodic intervals, which is closer than 19-tone
>>equal temperament.
>
>Come again?

7/4, 5/4, 3/2

>>[Reference Bobby McFerrin's "The Voice"]
>
>That's a great album -- which track are you thinking of?

Harmonic series segment like things can be heard throughout the album,
especially on track 7 in the "We're in the money" section.

>>You're assuming they approximate a utonal structure? Stuff I've heard is
>>usually closer to 7/6 or 19/16. That may be because of the linear series,
>>or the harmonic series, but in neither case are subharmonics implicated.
>>There is the exception of ethnic flutes, which often play subharmonic
>>series melodies, and after hearing such, it isn't too hard to sing them...
>
>Are you buying into the Schlesinger equally-spaced holes argument? No,
>flutes aren't quite like strings.

What do equal-spaced holes on flutes produce, then?

>I think they could arise on other instruments and even vocally, though; if
>the common overtone happens to coincide with a resonance, a subharmonic
>arpeggio would really stand out.

That's another thing -- perhaps subharmonic structures are important in
melody, assuming that first sentence of mine was true. But maybe it's true
not because of the virtual pitch mechanism, but because of the spectral
mechanism -- maybe the ear just hears just melodic intervals better because
it is aware of a common harmonic.

>By the way, Carl, you've referred to "linear series" twice now. Are you
>being Wilsonified?

I was Wilsonified long ago.

>>Historical evidence, maybe not (although Wilson may disagree).
>
>Any examples?

Not that I can remember, but I seem to recall him explaining an ethnic
scale as a chain of something other than 3/2's (was it one of the
permutations articles?). Kraig?

>>But certainly the melodic validity of Keenan's chain-of-minor-thirds scales
>> -- or the 22-out-of-41 scale you proposed -- is not in dispute,
>
>I recall much dispute over the latter from yourself, and please give us an
>mp3 of a nice melody with the former.

I disputed the scale only because it's so big -- most melodies would be
subsets of it. And due to perceptual limitations, even a melody that did
use the entire scale would probably fail to convey a 22-member melodic
pitch set. But it surely contains a wealth of great subsets.

Eeek! I've been having the worst time getting to my server today. My ISP
is periodically taken with loosing connection to whole portions of IP
space. At last, you may try the following, in the proper 8-tone subset of
Keenan's scale...

http://lumma.org/01.mp3 - 274K
http://lumma.org/02.mp3 - 634K
http://lumma.org/03.mp3 - 574K
http://lumma.org/04.mp3 - 1.02M

..or, go for all of them, plus 3 more at...

http://lumma.org/8all.zip - 3.37M

-Carl

>>I'd call +/-10 cents a fair estimate of the accuracy of singing the
>>"base-two under" 7-limit melodic intervals, which is closer than 19-tone
>>equal temperament.
>
>7:4 is 21 cents off in 19-tone equal temperament. And the point is?

If I said that 1/4 comma meantone comes within 6 cents, what would the
point be?

>>I was Wilsonified long ago.
>
>So tell me, why does he see modulus-n as having two primary representations,
>one as a linear series, and one as a high-limit diamond or CPS with added
>notes? What about representations with intermediate numbers of dimensions?

Why do I feel like I'm being set up? Extended reference requires a basic
set of intervals be available from several starting points. You can view
each representation as the most compact pitch set providing the highest
number of starting points as the basic set is varied. These are only
snapshots of what may happen in a performance -- the performer is likely to
mix and match. Is there another answer that contradicts something I've said?

>That was the point of it. Under those qualifications, it would be hard to
>dispute the melodic validity of any large scale with regularities.

True. And if that was the point of it, have you since changed your mind
about chains of 5/4's?

-C.